The document describes an OpenFOAM shape optimization example where the goal is to modify a geometry to match a target parabolic velocity profile at the outlet by running the adjoint solver. Over 20 iterations, the geometry is modified based on sensitivity maps from the adjoint method, with inward displacements where sensitivity is positive and outward displacements where sensitivity is negative. This improves the match of the calculated outlet velocity distribution to the target profile by 64%.

1. 1
Adjoint Shape Optimization Example #1
using OpenFOAM
- Target Velocity Profile at Outlet -
Fumiya Nozaki
Last Updated: 11 August 2014
English
Keywords：
• Continuous Adjoint
• Shape Optimization
• OpenFOAM

2. 2
Optimization objective
Inlet
Outlet
Fixed velocity:
Parabolic profile
Fixed pressure
Target velocity profile
@ outlet：
Parabolic profile
Change shape to obtain
the target velocity
profile @ outlet
• Laminar flow
• Air @ 20℃

3. 3
Objective function
𝐼 =
𝑐
2
𝒖 − 𝒖 𝑡𝑎𝑟𝑔𝑒𝑡
2
𝑑𝛤
𝑜𝑢𝑡𝑙𝑒𝑡
Difference between the calculated velocity and target velocity profile
Calculated by
simpleFoam
Target velocity
profile
Repeat shape modifications to get smaller value of 𝐼
Adjoint method gives
this information

4. 4
Sensitivity map
Positive sensitivity area
⇒ Inward surface displacement
improves the objective
Sensitivity distribution
Negative sensitivity area
⇒ Outward surface displacement
improves the objective

5. 5
Shape comparison between baseline and modified geometry
baseline modified
Shape
modifications
20 times

6. 6
Outlet velocity distribution
baseline modified
Improved by 64%

7. 7
Thank
You!